Fractal dimension for fractal structures: A Hausdorff approach

This paper provides a new model to compute the fractal dimension of a subset on a generalized-fractal space. Recall that fractal structures are a perfect place where a new definition of fractal dimension can be given, so we perform a suitable discretization of the Hausdorff theory of fractal dimension. We also find some connections between our definition and the classical ones and also with fractal dimensions I & II (see M.A. Sánchez-Granero and M. Fernández-Martínez (2010) [16]). Therefore, we generalize them and obtain an easy method in order to calculate the fractal dimension of strict self-similar sets which are not required to verify the open set condition.